Question: A radioactive material decays at a rate of change proportional to the current amount, $Q$, of the radioactive material. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $Q(t)=-kQ$ (Choice B) B $\dfrac{dQ}{dt}=-kQ$ (Choice C) C $Q(t)=-Q^{kt}$ (Choice D) D $\dfrac{dQ}{dt}=-Q^{kt}$
Solution: The amount of radioactive material is denoted by $Q$. The rate of change of the amount is represented by $Q'(t)$, or $\dfrac{dQ}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the current amount, $Q$, of radioactive material. We use a negative coefficient here to show that the change is decreasing the amount of radioactive material, so we have $-kQ$. In conclusion, the equation that describes this relationship is $\dfrac{dQ}{dt}=-kQ$.